Bias and sensitivity analyses for linear front-door models

The front-door model allows unbiased estimation of a total effect in the presence of unobserved confounding. This guarantee of unbiasedness hinges on a set of assumptions that can be violated in practice. We derive formulas that quantify the amount of bias for specific violations, and contrast them with bias that would be realized from a naive estimator of the effect. Some violations result in simple, monotonic increases in bias, while others lead to more complex bias, consisting of confounding bias, collider bias, and bias amplification. In some instances, these sources of bias can (partially) cancel each other out. We present ways to conduct sensitivity analyses for all violations, and provide code that performs sensitivity analyses for the linear front-door model. We finish with an applied example of the effect of math self-efficacy on educational achievement.